The exponential regime in which strong out-of-sample guarantees are imposed with \(a_T/T \rightarrow r\)  (\(a_T \sim rT\) ) for \(r>0\)  has been studied by [1]}. Here the decision-maker desires an exponentially decreasing probability of out-of-sample disappointment. They prove in a less general setting|in which the considered formulations do not directly depend on the data size|that the DRO predictor with Kullback-Leibler ball
\(\hat{c}(x,\hat{\mathbb {P}}_T,T) = \sup _{\mathbb {P}^{\prime }\in \mathbb {P}} \left\lbrace  c(x,\mathbb {P}^{\prime })\ : \ \sum _{i\in \Sigma }\hat{\mathbb {P}}_T(i) \log \left( \frac{\hat{\mathbb {P}}_T(i)}{\mathbb {P}^{\prime }(i)} \right)\le r \right\rbrace \) 
